Origins and demonstrations of electrons with orbital angular momentum

Abstract

The surprising message of Allen et al. (Allen et al. 1992 Phys. Rev. A 45, 8185 (doi:10.1103/PhysRevA.45.8185)) was that photons could possess orbital angular momentum in free space, which subsequently launched advancements in optical manipulation, microscopy, quantum optics, communications, many more fields. It has recently been shown that this result also applies to quantum mechanical wave functions describing massive particles (matter waves). This article discusses how electron wave functions can be imprinted with quantized phase vortices in analogous ways to twisted light, demonstrating that charged particles with non-zero rest mass can possess orbital angular momentum in free space. With Allen et al. as a bridge, connections are made between this recent work in electron vortex wave functions and much earlier works, extending a 175 year old tradition in matter wave vortices.This article is part of the themed issue 'Optical orbital angular momentum'.

Keywords: electron vortex; matter wave interferometry; optical angular momentum.

Figure 1.

Early work on optical diffraction (a,b) from two-dimensional masks is reproducible with electron diffraction (c,d). (a) An optical mask produced by Willis (adapted from [28]) simulates a natural crystal edge dislocation, with a topological charge of 1 in the x direction and 0 in the y direction. (b) Light diffracted by this mask produces optical vortices (figure adapted from [28]). Here the (0,3), (0,4) and (0,5) diffraction spots corresponding to optical vortices with ℓ=3,4 and 5, respectively, have been enlarged to reveal the central dark spots characteristic of topological charge. (c) A nanofabricated two-dimensional grating for electrons simulates an edge dislocation in a single crystal lattice plane. (d) The resulting electron diffraction pattern forms a two-dimensional array of electron vortices. In this diffraction image (300 keV, λ=1.97 pm), the diffraction spots were defocused in order to enlarge the diffraction spots.

Figure 2.

Schematic of electron vortices produced from a diffraction hologram. A spatially coherent electron wave function (a) illuminates a nanofabricated forked diffraction grating (b). The resulting diffracted portions of the wave function (c) possess quantized phase vortices. This diffraction pattern from many non-interacting electrons being diffracted is imaged in an electron microscope (d). For the experimental image adapted for (d), the undiffracted 0-order beam was blocked by a beam stop.

Figure 3.

Illustration of the creation of an electron vortex from the monopole-like magnetic field (black lines) of a ferromagnetic rod. The rod is in a plane perpendicular to the elecron optic axis, with one end placed on the optic axis. The field inside the needle strongly deviates from that of a monopole, but the material of the rod is sufficiently thick to scatter out of the optical system electrons that would be otherwise sensitive to the field there. Figure courtesy of Arthur Blackburn.

Figure 4.

(a) A TEM micrograph image of a silicon nitride forked diffraction grating used to imprint spiral phase onto electron beams. The lighter lines are grooves etched partially through the silicon nitride membrane using focused ion beam (FIB) milling, and have a spatial periodicity of 76 nm. (b) The fork in the diffraction grating encodes an azimuthally varying spatial phase (represented by colour scale) in the lateral registration of the grooves, revealed here by Fourier-filtering the TEM micrograph image in (a). (c) A radially averaged plot of the phase in (b) shows that it varies smoothly and linearly as a function of the polar angle ϕ and wraps continuously to 2π. In an actual electron diffraction experiment, this spiral phase is imprinted onto the +1 diffraction order, and the opposite winding direction is imprinted onto the −1 diffraction order.

Figure 5.

(a) A TEM image of the 80 keV electrons diffracted from the grating in figure 4. The central order has been removed by a beam block. A colour scale is used for this and subsequent images of diffraction patterns in order to make less intense features visible. (b–d) Magnified images of the +1, +2 and +3 diffraction orders. (e) Azimuthally averaged radial profiles of each of these diffracted beams reveals the dependence of the vortex core size on the topological charge. The dashed lines are fits derived from a Laguerre–Gaussian model of the electron wave functions (equation (4.1)).

Figure 6.

(a) An SEM micrograph image of a nanofabricated silicon nitride diffraction hologram with 15 extra grooves for producing electron beams with high OAM. (b) The ℓ=15 phase topology encoded in the hologram can be revealed by Fourier-filtering the SEM micrograph image. (c) TEM images of an actual electron diffraction pattern (300 keV) show electron vortex beams with multiples of 15 units of OAM.

Figure 7.

Electrons with extremely high OAM diffracted from a grating with topological charge of 200 (a) and 800 (b). A colour scale has been applied to (a) to make higher orders visible. For (b), the 0th-order beam has been removed by a beam block and a log scale applied to the image to make lower intensity, higher diffraction orders with extremely high amounts of OAM visible in the image. Electrons with 4000 are visible. Such beams are very sensitive to imperfections in the electron optics, and any small leftover astigmatism results in elliptic distortions seen here.

Figure 8.

Electron interferometry reveals the spiral phase topology imprinted onto electron vortex beams diffracted from the grating shown in figure 4. (a) An intermediate-field (far-defocus) TEM image of the grating reveals interference between the diffracted wave (circled in dashed yellow) and a reference wave directly transmitted through the surrounding transparent membrane, as discussed in [12]. The brighter concentric circular pattern on the right is the undiffracted 0-order of the grating. (b) Holographic phase reconstruction of the interferogram confirms the spiral phase of the diffracted beam.

Figure 9.

Transformation of electron OAM states within astigmatic electron optical systems. A magnetic quadrupole can be used to introduce the astigmatic perturbation to vortex beams, useful for mode conversion [62]. The central dark spot of a single topological charge state distorts into an edge-type phase dislocation, forming a dark line across the spot (a). (c) A series of images of astigmatic higher OAM states shows that each state becomes an ellipse. The numbers in each image of this sequence denote the astigmatism (expressed in scaled instrumental units) introduced by tuning the field gradient of a magnetic quadrupole lens (the diffraction stigmator on the TEM). One can see that negative OAM states on the left side of the diffraction pattern are all tilted in different directions from those on the right. The evolution of each OAM state is reminiscent of a current loop precessing in space. At high enough quadrupole excitation (lower rightmost image), the OAM states seem to have inverted.

Figure 10.

Representations of an electron vortex wavepacket. (a) A Laguerre–Gaussian wave function serves as a model for electron de Broglie wavepackets in a focused electron vortex beam. The helical wavefront is plotted as a surface of constant phase, with an opacity is proportional to |ψ_ℓ(ρ,ϕ,z)|², the probability distribution of the wave function. (b) A geometric model of the beam can be derived by evaluating the local momentum of the Laguerre–Gaussian function. Here, the probability current evaluated at the radius of peak amplitude of the wave function forms a collection of skewed straight rays. (c) In the centre-of-mass (CM) frame of reference, the circulating electron vortex probability distribution is torus-shaped, depicted here as a surface of constant probability. The quantized OAM, L_z, about the toroid centre (black arrow) combined with the electron’s charge gives rise to an associated magnetic moment μ_ℓ (red arrow) described by equation (4.2). In this CM frame, both the transverse width of the doughnut, determined by the width of the beam, and the longitudinal extent of the torus along z, related to the spread in longitudinal momentum, evolve in time.